In the final verses of Galatians 2, Paul made the argument that everyone involved in this controversy in Galatia (himself, the teachers who oppose him, Peter and the Jerusalem council whose authority has probably been invoked by the teachers) knows that no one is made righteous by works of the Law but by the faithfulness of Jesus Christ. Paul says that since this is true it would be absurd to force the Galatians to observe these works of the Law which the teachers are telling them they must observe. In fact, to do so would be to nullify the grace of God which came through the faithfulness of Jesus. This is the main point of Paul's letter to the Galatians. In 3:1-14, Paul begins to make his supporting arguments for this main point.
Paul's first supporting argument begins with the experience of the Galatians themselves. He says in v.2 "This is the only thing I want to find out from you: did you receive the Spirit by the works of the Law or by the message of (Christ's) faithfulness?" Like a good lawyer, Paul only asks questions for which he knows the answer. Paul's whole argument here rests on the Galatians agreeing that they received the Spirit when they heard about the faithfulness of Christ, long before these teachers ever showed up to tell them about works of the Law. (This is an interesting point in itself since the Galatians' reception of the Spirit must have been such an obvious and demonstrable thing that no one would have debated whether or not they had indeed received it. It is such a sure and unquestionable experience that Paul could build his argument on it. Perhaps we are to imagine something happened in Galatia similar to the story of Cornelius and his household in Acts 10?) For Paul, this alone should be enough to settle the matter. If the Galatians received the Spirit before being taught about works of the Law, then clearly God has already accepted them and began to make them righteous and these works of the Law are an unnecessary addendum to what God is already doing in them.
Paul drives this point home further with an argument from Scripture. Paul says it was the same way with Abraham as it is with the Galatians. Abraham placed his trust in God by faithfully following God's calling for him and God said that Abraham was righteous for doing so long before God ever commanded Abraham to be circumcised. Like the Galatians, Abraham was counted as righteous by God before "works of the Law" were even a part of the picture. Therefore, Abraham exemplifies Paul's argument that it is those who live faithfully whom God makes righteous, not those who seek to find life in circumcision, food laws, and sabbath observance. Paul says that the Scripture even anticipates the inclusion of the Gentiles in this righteous living through faithfulness when it told Abraham that all the nations would be blessed through him; not just Jews but Gentiles as well.
Having made his parallel between Abraham and the Galatians, Paul now turns his argument to demonstrating why works of the Law are not a part of the righteousness which comes through faith/faithfulness (and in doing so writes four of the most theologically dense and widely debated verses in the New Testament). Paul begins this point by saying that everyone who is of the works of the Law are under a curse and he quotes Deuteronomy to support this point; "cursed is everyone who does not keep everything written in the book of the Law to do them." Some take these words to mean that everyone who tries to follow the Law is cursed because following the Law is impossible but this is not at all what Paul is saying. Jews did not find the Law impossible to follow. Paul declared himself blameless as to the Law (Phil 3:5). Paul is simply saying that anyone who chooses to derive their identity from the works of the Law stands under the threat of a curse; they may not actually be cursed if they keep the whole Law but the threat is always there. (If you read in some of Paul's writing in Romans at this point, then you can obverse that in Paul's mind, this is all the Law can really do: pronounce curses. It was supposed to be a life giving blessing that would protect Israel from sin. Instead, it was powerless to prevent sin and itself became a tool of sin. Therefore, all the Law could do was threaten, it could not make one alive, make one righteous.)
In contrast to the "Law option" in which everyone stands under the threat of a curse, Paul argues that there is now another option: that of faith/faithfulness; and, in fact, this faith/faithfulness option is the only true option for being made righteous. In v.11, Paul says "since it is evident that no one is made righteous before God by the Law, the righteous will live by faith/faithfulness." It is evident that no one is made righteous by the Law because of Israel's history. Israel had followed the Law for hundreds of years but it had never made Israel truly righteous and Israel knew it. Israel's own prophets described their longing for the day when God would pour out his Spirit to truly make them righteous in a way that the Law had not. Since the Law failed to make Israel righteous, Paul argues there must be another option for righteousness and that option is the faith/faithfulness in which the Galatians were already living before these teachers showed up telling them that they must follow the works of the Law.
In v.12, Paul says these "works of the Law are not of faith but those who do them will live by them." In other words, life by works of the law and life by faith/faithfulness can not coexist, they are incompatible. This is because they both represent an ultimate orientation in life. That is the point Paul is making by quoting Leviticus 18:5: "those who do these things live by them"; that is, those who follow these works of the Law live their whole life by them. The problem with works of the Law is not the works themselves. The problem is orienting one's whole life around them, expecting to find life in them, attempting to be made righteous by them when instead one should be seeking life and righteousness in the faithfulness of Christ and orienting one's life around faith in him.
After all, it is not these works of the Law which redeemed us from the curse of the Law. It is Christ who did that by becoming a curse for us. Christ became a curse by being crucified. Without the resurrection, crucifixion was a certain and public sign that Jesus was not the messiah he had claimed to be. Furthermore, Paul points out in quoting Deuteronomy 27:26 that anyone whose dead body was left hanging for public displayed was considered cursed by the standards of the Law. However, Christ willingly took that curse upon himself so that we would no longer have to stand under the threat of the Law's curse.
In v.14, Paul says that Christ became a curse for us "in order that the promise of Abraham might come to the gentiles through Christ Jesus in order that we might receive the promise of the Spirit through faith/faithfulness". That is, it is Christ's death which opens the blessing of Abraham and the pouring out of the Holy Spirit to gentiles as well as Jews and that is essentially what it means to be redeemed from the curse of the Law. We have been redeemed, bought back, freed from the slavery of the threat of the Law's curse because there is now another option for righteousness besides living under the Law and its threat; the option of living in the faithfulness of Jesus Christ. And, of course, Paul would say this new option is the only real option since the old option of living under the Law was really powerless to protect again sin and make one righteous anyway.